Heat 2 tablespoons of the olive oil in a large heavy skillet over medium-high heat. Add the onions, bell peppers, celery, salt, and cayenne and cook, stirring occasionally, until the vegetables are tender, about 5 minutes. Add the green onions and parsley and cook just until wilted, about 1 minute. Remove from the heat.
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Place the crabmeat in a large bowl and add the cooked vegetables. Add the mayonnaise, cheese, and egg and stir gently with a large wooden spoon. Add the bread crumbs and stir gently, being careful not to break up the crabmeat too much. Form the mixture into twelve 2-inch-diameter cakes.
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Combine the flour and Essence in a medium bowl. Lightly coat both sides of each cake with the flour mixture. Heat the remaining 2 tablespoons olive oil in a large heavy skillet over medium-high heat. In batches, without crowding, cook the cakes until golden brown, about 3 minutes per side, adding more oil as needed. Transfer to paper towels to drain.
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To serve, place two crab cakes on each plate, top each cake with 1 poached egg, and spoon about 1/3 cup of the sauce over each serving.
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Yield: 6 servings